Gaussian Beam Calculator for Creating Coherent Sources INTRODUCTION Coherent sources are represented in FRED using a superposition of Gaussian beamlets. The ray grid spacing of the source is used to determine the waist and divergence of the beamlets. To properly model coherent propagation properties, the beamlets must remain Gaussian and perform best when they obey the paraxial approximation. As a result, some Gaussian beam calculations should be performed to determine the appropriate grid size and number of rays for a coherent source. The attached script uses desired source parameters to calculate and display the proper inputs for a new coherent source. This calculator is valid for the following source types: Simplified Source o Collimated Source (plane wave) o Laser Beam (Gaussian 00 mode) Detailed Source o Grid Plane (rectangular array of points arranged on a plane) o Hexagonal Plane (hexagonal array of points arranged on a plane) Figure 1. Gaussian beam calculator dialog
COHERENT SOURCES AND BEAMLETS A note on terminology: the terms beam waist radius and beam far-field divergence refer to the overall Gaussian beam that is being modeled. Beamlet waist radius and beamlet far-field divergence refer to the individual Gaussian beamlets that FRED creates to represent the overall beam. There is a beamlet for every base ray. The beamlets are described and propagated using rays. A central base ray represents the trajectory of the beamlet (its position and direction is analogous to the corresponding ray of a non-coherent source) and additional secondary waist and divergence rays track the evolution of the beamlet parameters. The relationship between a beamlet and its corresponding rays is shown in the figure below. The rays fully describe the beamlet characteristics as they undergo refraction, reflection, and diffraction. This process is referred to as complex raytracing. At any plane in the system, the coherent field can be calculated by determining the contribution of each beamlet at each position on the analysis surface and accounting for the phases. Figure 2. Gaussian beamlet with corresponding rays When a coherent source grid is defined, the beamlet waist radius becomes,,,, 4 π (1) where, is the semi-aperture, is the overlap factor, and, is the number of rays across the full aperture. These parameters may differ in the x and y directions for rectangular and elliptical ray grids. The overlap factor is the fractional overlap between adjacent beamlets on the grid, and has a default value of 1.5 (which should rarely be
changed). The 4 factor arises from the fact that FRED actually uses the / point for creating secondary rays along the beamlet waist, while the more common waist radius definition is used here. The divergence,, of the beamlet follows the relation for Gaussian beams, tan where is the index of refraction, and is the wavelength. (2) In order for the model to be accurate, the beamlets must remain Gaussian and perform best when they obey the paraxial approximation. This is perhaps the most important consideration in properly implementing coherent raytracing. Attempting to operate outside this paraxial limit negates the ability of Gaussian beamlets to accurately sample optical components as they propagate. Failure of secondary rays to remain wellcorrelated with their base ray may lead to coherent ray errors and erroneous irradiance calculations. While not precisely defined, the paraxial approximation can be expressed in many forms, including tanθ θ and θ << π. In both cases, a reasonable choice for θ is 0.1 radians, or about 6 degrees. Plugging this into Equation 2 and assuming propagation in air ( 1) yields the result that the beamlet waist radius ω o must be greater than or equal to about 3λ. In practice, the user should consider operating with some amount of margin, maybe 5-10λ. If the desired number of rays is known, this beam waist radius restriction can be plugged into Equation 1 and solved for the appropriate grid semiaperture,,. This is precisely what the Gaussian Beam Calculator script does. USING THE CALCULATOR Figure 3 shows the calculator dialog with parameter dependencies. Each input parameter was assigned a differently colored circle. Colored squares have been placed next to each calculated value that is dependent on the input parameter of the corresponding color. For example, the number of rays across the full x/y aperture depends on the total number of rays and the grid shape. Note that if the beamlet waist restriction is defined in terms of waves, changing the wavelength affects the beamlet waist and thus also all the calculated values that are dependent on the beamlet waist restriction.
Figure 3. Calculator dialog with parameter dependencies shown. When an input parameter represented by a colored circle is changed, it affects all the calculated values that include the same colored square. The calculator includes a default value for each input parameter. These provide a convenient starting point in case of uncertainty. Pressing the Calculate button updates the derived values in the Calculated values box. Pressing Print Results displays a report containing the input parameters and calculated values in the Command Window. When either Calculate or Print Results is pressed, the input values are retained and the dialog remains open. Pressing the Close button closes the dialog and terminates the script. All of the four supported source types listed in the Introduction use a combination of the first three parameters in the Calculated values box as inputs. The beamlet and beam far field-divergences are calculated for reference. The following example demonstrates the use of this calculator. Let s say we want to model a 1 mm-diameter TEM00 laser beam at 0.5 um in air, defined at its waist. We determine that we d like to trace at least 50,000 rays to accurately sample the system, but less than 250,000 to keep the raytrace fast. We decide to use the simplified source type Laser Beam (Gaussian 00 mode), which requires three input parameters: the beam waist semi-aperture, the semi-aperture of the sample plane at the waist, and the number of sample points across the sample plane. How do we determine what these
last two values should be to accurately model the laser and ensure that the beamlets FRED generates are paraxial? This is where the calculator comes in. Going down the lefthand side of the calculator inputs, we put in 0.5 for the waist radius, 0.5 for the wavelength, 250000 for the number of rays, and leave the defaults of n=1, a circular grid, 1.5 overlap factor, and 10 waves for the beamlet waist restriction. Pressing Calculate displays the parameters. The grid semi-aperture is 1.67 mm, which is a bit large for a 0.5 mm beam radius. To reduce the grid semi-aperture to around 1 mm, we can either decrease the beamlet waist restriction to 7 waves (which increases the beamlet divergence to 2.6 degrees), or reduce the number of total rays to 100,000 (which keeps the beamlet divergence at 1.82 degrees). Both of these are valid options. The decision as to which to use depends on whether a larger number of rays or a smaller beamlet divergence is more important. For example, a small beamlet divergence is desirable if the beam travels over a long distance. Such trade-offs are common when defining coherent sources and the GB calculator provides a convenient way to quickly determine the effect various parameters have on the source definition. If an elliptical or rectangular beam with unequal parameters in the x and y directions is being defined, the two sets of parameters may be determined separately by using the calculator twice. If a source type other than Laser Beam (Gaussian 00 mode) is used, the intensity profile (e.g. Gaussian) should be specified under the Position Apodization in the Power tab of the Detailed Source edit/create dialog. If no apodization is defined, the beam width will be equal to the source grid aperture. Although the beam waist of the source is one of the inputs, the calculator can be used for non-gaussian coherent beams. As shown in Figure 3, the beam waist is only used to calculate the beam divergence angle and does not affect any of the source input parameters.
EQUATIONS USED IN CALCULATOR Table 1. Input parameter variables Input parameter label Waist radius at e^-2 (mm) Wavelength (um) Total number of rays Index of refraction Grid shape Overlap factor Beamlet waist restriction (um) Variable used in equations, Circ or Rect, or, Number of rays across the full aperture ( ) If grid shape = Rect: N (3) If grid shape = Circ: X/Y Semi-aperture of grid in mm (): 2 N π (4), 1000 π 4 (5) Semi-aperture of 1 hexagonal cell in mm ( ): Beamlet far-field divergence ( ): 2 (6) tan, (7) Beam far-field divergence ( ): tan, (8)
THE SCRIPT The script is structured as follows: 1. Define variables for calculated parameters and set them as blank strings. 2. Start while loop that runs as long as the user presses Calculate or Print Results in the Dialog. 3. Define dialog, using variables for the string parameters displayed in the Calculated values box. 4. Put in default values if the dialog is being displayed for the first time. 5. Display the dialog and assign all input values to variables. 6. Check if any of the input variables equal 0. a. Yes: assign NA to all calculated parameters. b. No: calculate all dependent parameters in turn and convert them to strings to be displayed in the Dialog. 7. Print report in the Output Window if the Print Results button was pressed. 8. Go to while loop beginning and repeat Steps 2-8 if Calculate or Print Results was pressed. Exit loop and script otherwise (i.e. Close button was pressed).